Method for calculating parameter changing domain of loads under a case that guarantees constant locational marginal price in electricity market

ABSTRACT

The disclosure provides a method for calculating a parameter changing domain of loads under a case that guarantees a constant locational marginal price in an electricity market, which relates to the electricity market field of the power system. With the method in the disclosure, the clearing model on the locational marginal price in the general form is established, and the safe changing domain of the locational marginal price with respect to the loads may be derived and calculated based on the first-order KKT condition expansion of the clearing model on the locational marginal price in the general form. The method may quickly calculate the parameter changing domain of different nodal loads under the condition of a clear base state in the current power market. When the increment of the nodal loads is subordinate to the changing domain, the locational marginal price may remain unchanged.

TECHNICAL FIELD

The disclosure relates to the electricity market field of the powersystem, and more particularly to a method for calculating a parameterchanging domain of loads under a case that guarantees a constantlocational marginal price in an electricity market.

BACKGROUND

With the continuous development of electricity market theory andapplication, an economic dispatch model based on optimal power flow hasbeen widely applied in the electricity market clearing. The locationalmarginal price based on the economic dispatch model has become amainstream tool in the electricity market. However, with the increasingpenetration rate of renewable energy access and the popularization ofelectric vehicles, the power system is increasingly likely to operateunder extreme conditions, causing system congestion and making thelocational marginal price vulnerable to various factors, especially tonodal loads. In this context, it is greatly significant to the safe andeconomic operation of the power system that the impact of the nodalloads on the locational marginal price is evaluated and a parameterchanging domain of loads is determined under an operation base state toguarantee a constant locational marginal price. It is the basis formarket information risk identification, market power analysis,congestion management, and adjustment of operation modes.

SUMMARY

The disclosure aims to provide a method for calculating a parameterchanging domain of loads under a case that guarantees a constantlocational marginal price in an electricity market, based on alocational marginal price clearing model in a current electricitymarket. The parameter changing domain of different nodal loads under aclearing base state in the current electricity market may be calculatedquickly. When an increment of the nodal loads is subordinate to thechanging domain, the locational marginal price may remain unchanged.

The method for calculating the parameter changing domain of loads in thepower system under the case that guarantees the constant locationalmarginal price in the electricity market, provided in the disclosure,may include the following.

(1) A clearing model on the locational marginal price in a general formis established, which has the following specific process.

(1-1) The clearing model on the locational marginal price is establishedas follows:

$\mspace{20mu}{\min\limits_{p_{i}}{\sum\limits_{i = 1}^{N_{g}}{c_{i}p_{i}}}}$$\mspace{20mu}{{{satisfying}:{\sum\limits_{i = 1}^{N_{g}}p_{i}}} = {\sum\limits_{j = 1}^{N}D_{j}}}$${P_{i}^{\min} \leq p_{i} \leq P_{i}^{\max}},{\forall{i \in {{N_{g} - F_{l}^{\max}} \leq {{\sum\limits_{i = 1}^{N_{g}}{S_{l,i}p_{i}}} - {\sum\limits_{j = 1}^{N}{S_{l.j}D_{j}}}} \leq F_{l}^{\max}}}},{\forall{l \in L}},$

where,

N_(g) represents a number of generator nodes in the power system;

N_(g) represents a set of serial numbers of the generator nodes, N_(g)□{1, 2, . . . , N_(g)};

N represents a total number of nodes in the power system;

L represents a set of serial numbers of branches in the power system,L□{1,2, . . . , L}, L represents a total number of the branches in thepower system;

p_(i),i∈N_(g) represents a power variable of generator i;

c_(i),i∈N_(g) represents a power cost coefficient of generator i;

P_(i) ^(max) represents an upper power limit of the generator node;

P_(i) ^(min) represents a lower power limit of the generator node;

D_(j),j∈{1,2, . . . , N} represents a nodal load of the power system;

F_(l) ^(max),l∈L represents a capacity of branch l; and

S_(l,i),l∈L, i∈{1, 2, . . . , N} represents a power transferdistribution factor.

There is the following linear relationship between locational marginalprices and Lagrangian multipliers of constraints:

Λ=τ−S _(LN) ^(T)(μ^(L,max)−μ^(L,min)),

where,

Λ represents a column vector including locational marginal prices of thenodes in the power system in an order of serial numbers of the nodes;

τ represents a Lagrangian multiplier for the power balance constraint

${{\sum\limits_{i = 1}^{Ng}p_{i}} = {\sum\limits_{i = 1}^{N}D_{i}}};$

μ^(L,max) represents a column vector including Lagrangian multipliersfor the upper bound constraint

${{\sum\limits_{i = 1}^{Ng}{S_{l.i}p_{i}}} - {\sum\limits_{i = 1}^{N}{S_{l.i}D_{i}}}} \leq F_{l}^{\max}$

in the power system in an order of serial numbers of the branches;

μ^(L,min) represents a column vector including Lagrangian multipliersfor the lower bound constraint

${- F_{l}^{\max}} \leq {{\sum\limits_{i = 1}^{Ng}\;{S_{l,i}p_{i}}} - {\sum\limits_{i = 1}^{N}\;{S_{l,i}D_{i}}}}$

in the power system in an order of serial numbers of the branches;

S_(LN) represents a power transfer distribution factor matrix; and

T represents a matrix transpose.

(1-2) Let a variable p_(i)′=p_(i)−P_(i) ^(min), to transform a decisionvariable p_(i) of the clearing model into a pure non-negative variablep_(i)′, and slack variables p_(i) ^(sl),f_(l) ^(sl,min), f_(l) ^(sl,max)are introduced to transform the clearing model into a linear programmingin a general form as follows:

$\min\limits_{p_{i}}{\sum\limits_{i = 1}^{Ng}\;{c_{i}p_{i}^{\prime}}}$${{satisfying}:{\sum\limits_{i = 1}^{Ng}\; p_{i}^{\prime}}} = {{\sum\limits_{i = 1}^{N}\; D_{i}} - {\sum\limits_{i = 1}^{Ng}\; P_{i}^{\min}}}$${{p_{i}^{\prime} + p_{i}^{sl}} = {P_{i}^{\max} - P_{i}^{\min}}},{{\forall{i \in {N_{g} - F_{i}^{\max}}}} = {{\sum\limits_{i = 1}^{Ng}\;{S_{l,i}p_{i}^{\prime}}} + {\sum\limits_{i = 1}^{Ng}\;{S_{l,i}P_{i}^{\min}}} - {\sum\limits_{i = 1}^{N}\;{S_{l,i}D_{i}}} - f_{l}^{{sl},\min}}},{\forall{l \in L}}$${{{\sum\limits_{i = 1}^{Ng}\;{S_{l,i}p_{i}^{\prime}}} + {\sum\limits_{i = 1}^{Ng}\;{S_{l,i}P_{i}^{\min}}} - {\sum\limits_{i = 1}^{N}\;{S_{l,i}D_{i}}} + f_{l}^{{sl},\max}} = F_{l}^{\max}},{\forall{l \in L}}$p_(i)^(′), p_(i)^(sl), f_(l)^(sl, min ), f_(l)^(sl, max ) ≥ 0.

(1-3) The clearing model obtained in (1-2) is simplified into the linearprogramming in the general form, as follows:

$\min\limits_{x}{\sum\;{c \cdot x}}$ satisfying : A ⋅ x = b, x ≥ 0

where,

matrix A and vectors c,b correspond to parameters of the clearing modelas follows:

${A = \begin{bmatrix}e_{G} & \; & \; & \; \\I_{G} & I_{G} & \; & \; \\S_{LG} & \; & {- I_{L}} & \; \\S_{LG} & \; & \; & I_{L}\end{bmatrix}},{x = \begin{bmatrix}p^{\prime} \\p^{sl} \\f^{{sl},\min} \\f^{{sl},\max}\end{bmatrix}}\;,{b = \begin{bmatrix}{{e_{D}D} - {e_{G}P^{\min}}} \\{P^{\max} - P^{\min}} \\{{S_{LN}D} - {S_{LG}P^{\min}} - F^{\max}} \\{{S_{LN}D} - {S_{LG}P^{\min}} + F^{\max}}\end{bmatrix}},{c = \left\lbrack {c^{T}\mspace{14mu} 0\mspace{14mu} 0\mspace{14mu} 0} \right\rbrack}$

where,

e_(G) is a matrix whose elements of dimension 1×N are all 1;

e_(D) is a matrix whose elements of dimension 1×N are all 1;

I_(G) is a unit matrix with dimension N_(g)×N_(g);

I_(L) is a unit matrix with dimension L×L;

S_(LG) is a sub-matrix formed by columns corresponding to the generatornodes.

(2) The parameter changing domain of loads under the case thatguarantees the constant locational marginal price is derived andcalculated based on a first-order KKT condition of the clearing model in(1-3) in the general form.

(2-1) The first-order KKT condition in an incremental form may bederived as follows:

$\left\{ {\begin{matrix}{{A \cdot x^{*}} = b} \\{x^{*} \geq 0} \\{{{{A^{T} \cdot \omega} + r} = c^{T}},{r \geq 0}} \\{{r^{T} \cdot x^{*}} = 0}\end{matrix},} \right.$

where,

ω is a Lagrangian multiplier vector of the constraint condition A·x=b;

r is a Lagrangian multiplier vector of the constraint condition x≥0;

c,b are independent variables in the KKT condition;

ω,r,x* are dependent variables in the KKT condition.

It is supposed that in a base state c=c₀ and b=b₀, the dependentvariables in the KKT condition may be ω=ω₀,r=ω₀,x*=x*₀.

In order to ensure that the dependent variables ω,r remain unchangedafter the independent variable b is superimposed by □b, it is necessaryto ensure that when the independent variables become c=c₀ and b=b₀+□b,the dependent variables in the KKT condition satisfy the following form:ω=ω₀, r=r₀, x*=x₀*+□x*.

Therefore, in the base state c=c₀, b=b₀, the KKT condition is asfollows:

$\left\{ {\begin{matrix}{{A \cdot x_{0}^{*}} = b_{0}} \\{x_{0}^{*} \geq 0} \\{{{{A^{T} \cdot \omega_{0}} + r_{0}} = c^{T}},{r \geq 0}} \\{{r_{0}^{T} \cdot x_{0}^{*}} = 0}\end{matrix}.} \right.$

When the independent variables change in the base state, the KKTcondition is as follows:

$\left\{ {\begin{matrix}{{A \cdot \left( {x_{0}^{*} + {\bullet\; x^{*}}} \right)} = {b_{0} + {\bullet\; b}}} \\{{x_{0}^{*} + {\bullet\; x^{*}}} \geq 0} \\{{{{A^{T} \cdot \omega_{0}} + r_{0}} = c^{T}},{r_{0} \geq 0}} \\{{r_{0}^{T}\left( {x_{0}^{*} + {\bullet\; x^{*}}} \right)} = 0}\end{matrix}.} \right.$

The expansion equation of the first-order KKT condition in theincremental form may be derived from the above equation as follows:

$\left\{ {\begin{matrix}{{{A \cdot \bullet}\; x^{*}} = {\bullet\; b}} \\{{{r_{0}^{T} \cdot \bullet}\; x^{*}} = 0} \\{{x_{0} + {\bullet\; x^{*}}} \geq 0}\end{matrix}.} \right.$

(2-2) A projection matrix is designed to derive the parameter changingdomain of loads under the case that guarantees the constant locationalmarginal price:

The projection matrix P of an equation r₀ ^(T)·□x*=0 is defined asfollows:

P=I−r ₀·(r ₀ ^(T) ·r ₀)⁻¹ ·r ₀ ^(T).

The expansion equation of the first-order KKT condition in theincremental form may be transformed into the parameter changing domainof loads under the case that guarantees the constant locational marginalprice as follows:

S□{A·P·□y|x*+P□y≥0, P=I−r ₀·(r ₀ ^(T) ·r ₀)⁻¹ ·r ₀ ^(T) , □y∈R ^(n)}.

The method for calculating the parameter changing domain of loads in thepower system under the case that guarantees the constant locationalmarginal price in the electricity market, provided in the disclosure,may have the following advantages.

With the method for calculating the parameter changing domain of loadsin the power system under the case that guarantees the constantlocational marginal price in the electricity market, the clearing modelon the locational marginal price in the general form is established, andthe safe changing domain of the locational marginal price with respectto the loads may be derived and calculated based on the first-order KKTcondition expansion of the clearing model on the locational marginalprice in the general form. The method may quickly calculate theparameter changing domain of different nodal loads under the conditionof a clear base state in the current power market. When the increment ofthe nodal loads is subordinate to the changing domain, the locationalmarginal price may remain unchanged. The parameter changing domain ofloads in the power system, calculated by the method of the disclosuremay be used for the comprehensive evaluation of power market clearingresults and assisting the operation of the power market.

DETAILED DESCRIPTION

The method for calculating the parameter changing domain of loads in thepower system under the case that guarantees the constant locationalmarginal price in the electricity market, provided in the disclosure,may include the following.

(1) A clearing model on the locational marginal price in a general formis established, which has the following specific process.

(1-1) The clearing model on the locational marginal price is establishedas follows:

$\min\limits_{p_{i}}{\sum\limits_{i = 1}^{Ng}\;{c_{i}p_{i}}}$${{satisfying}:{\sum\limits_{i = 1}^{Ng}\; p_{i}}} = {\sum\limits_{j = 1}^{N}\; D_{j}}$${P_{i}^{\min} \leq p_{i} \leq P_{i}^{\max}},{\forall{i \in {{N_{g} - F_{l}^{{ma}x}} \leq {{\sum\limits_{i = 1}^{Ng}\;{S_{l,i}p_{i}}} - {\sum\limits_{j = 1}^{N}\;{S_{l,j}D_{j}}}} \leq F_{l}^{\max}}}},{\forall{l \in L}},$

where,

N_(g) represents a number of generator nodes in the power system;

N_(g) represents a set of serial numbers of the generator nodes, N_(g)D{1, 2, . . . , N_(g)};

N represents a total number of nodes in the power system;

L represents a set of serial numbers of branches in the power system, LD {1,2, . . . , L}, L represents a total number of the branches in thepower system;

p_(i), i∈N_(g) represents a power variable of generator i andc_(i),i∈N_(g) represents a power cost coefficient of generator i, whichare declared and confirmed by the main body of each generator to therelevant power agencies.

P_(i) ^(max) represents an upper power limit of the generator node;

P_(i) ^(min) represents a lower power limit of the generator node;

D_(j),j∈{1,2, . . . , N} represents a nodal load of the power system;

F_(l) ^(max),l∈L represents a capacity of branch l; and

S_(l,i),l∈L,i∈{1, 2, . . . , N} represents a power transfer distributionfactor, which is calculated by relevant power agencies and released uponapplication.

There is the following linear relationship between locational marginalprices and Lagrangian multipliers of constraints:

Λ=τ−S _(LN) ^(T)(μ^(L,max)−μ^(L,min)),

where,

Λ represents a column vector including locational marginal prices of thenodes in the power system in an order of serial numbers of the nodes;

τ represents a Lagrangian multiplier for the power balance constraint

${{\sum\limits_{i = 1}^{Ng}\; p_{i}} = {\sum\limits_{i = 1}^{N}\; D_{i}}};$

μ^(L,max) represents a column vector including Lagrangian multipliersfor the upper bound constraint

${{\sum\limits_{i = 1}^{Ng}\;{S_{l,i}p_{i}}} - {\sum\limits_{i = 1}^{N}\;{S_{l,i}D_{i}}}} \leq F_{l}^{\max}$

in the power system in an order of serial numbers of the branches;

μ^(L,min) represents a column vector including Lagrangian multipliersfor the lower bound constraint

${- F_{l}^{\max}} \leq {{\sum\limits_{i = 1}^{Ng}\;{S_{l,i}p_{i}}} - {\sum\limits_{i = 1}^{N}\;{S_{l,i}D_{i}}}}$

in the power system in an order of serial numbers of the branches;

S_(LN) represents a power transfer distribution factor matrix; and

T represents a matrix transpose.

(1-2) Let a variable p_(i)′=p_(i)−P_(i) ^(min), to transform a decisionvariable p_(i) of the clearing model into a pure non-negative variablep_(i)′, and slack variables p_(i) ^(sl),f_(l) ^(sl,min), f_(l) ^(sl,max)are introduced to transform the clearing model into a linear programmingin a general form as follows:

$\min\limits_{p_{i}}{\sum\limits_{i = 1}^{Ng}\;{c_{i}p_{i}^{\prime}}}$${{satisfying}:{\sum\limits_{i = 1}^{Ng}\; p_{i}^{\prime}}} = {{\sum\limits_{i = 1}^{N}\; D_{i}} - {\sum\limits_{i = 1}^{Ng}\; P_{i}^{\min}}}$${{p_{i}^{\prime} + p_{i}^{sl}} = {P_{i}^{\max} - P_{i}^{\min}}},{{\forall{i \in {N_{g} - F_{i}^{\max}}}} = {{\sum\limits_{i = 1}^{Ng}\;{S_{l,i}p_{i}^{\prime}}} + {\sum\limits_{i = 1}^{Ng}\;{S_{l,i}P_{i}^{\min}}} - {\sum\limits_{i = 1}^{N}\;{S_{l,i}D_{i}}} - f_{l}^{{sl},\min}}},{\forall{l \in L}}$${{{\sum\limits_{i = 1}^{Ng}\;{S_{l,i}p_{i}^{\prime}}} + {\sum\limits_{i = 1}^{Ng}\;{S_{l,i}P_{i}^{\min}}} - {\sum\limits_{i = 1}^{N}\;{S_{l,i}D_{i}}} + f_{l}^{{sl},\max}} = F_{l}^{\max}},{\forall{l \in L}}$p_(i)^(′), p_(i)^(sl), f_(l)^(sl, min ), f_(l)^(sl, max ) ≥ 0.

(1-3) The clearing model obtained in (1-2) is simplified into the linearprogramming in the general form, as follows:

$\min\limits_{x}{\sum\;{c \cdot x}}$ satisfying : A ⋅ x = b, x ≥ 0

where,

matrix A and vectors c,b correspond to parameters of the clearing modelas follows:

${A = \begin{bmatrix}e_{G} & \; & \; & \; \\I_{G} & I_{G} & \; & \; \\S_{LG} & \; & {- I_{L}} & \; \\S_{LG} & \; & \; & I_{L}\end{bmatrix}},{x = \begin{bmatrix}p^{\prime} \\p^{sl} \\f^{{sl},\min} \\f^{{sl},\max}\end{bmatrix}}\;,{b = \begin{bmatrix}{{e_{D}D} - {e_{G}P^{\min}}} \\{P^{\max} - P^{\min}} \\{{S_{LN}D} - {S_{LG}P^{\min}} - F^{\max}} \\{{S_{LN}D} - {S_{LG}P^{\min}} + F^{\max}}\end{bmatrix}},{c = \left\lbrack {c^{T}\mspace{14mu} 0\mspace{14mu} 0\mspace{14mu} 0} \right\rbrack}$

where,

e_(G) is a matrix whose elements of dimension 1×N_(g) are all 1;

e_(D) is a matrix whose elements of dimension 1×N are all 1;

I_(G) is a unit matrix with dimension N_(g)×N_(g);

I_(L) is a unit matrix with dimension L×L;

S_(LG) is a sub-matrix formed by columns corresponding to the generatornodes.

(2) The parameter changing domain of loads under the case thatguarantees the constant locational marginal price is derived andcalculated based on a first-order KKT (Karush-Kuhn-Tucker) condition ofthe clearing model in (1-3) in the general form, which may include thefollowing.

(2-1) The first-order KKT condition in an incremental form may bederived as follows:

$\quad{\left\{ \begin{matrix}{{A \cdot x^{*}} = b} \\{x^{*} \geq 0} \\{{{{A^{T} \cdot \omega} + r} = c^{T}},{r \geq 0}} \\{{r^{T} \cdot x^{*}} = 0}\end{matrix} \right.,}$

where,

ω is a Lagrangian multiplier vector of the constraint condition A·x=b;

r is a Lagrangian multiplier vector of the constraint condition x≥0;

c,b are independent variables in the KKT condition;

ω,r,x* are dependent variables in the KKT condition.

According to the definition of the locational marginal price in thepower system, it may be seen from (1-1) that the relationship betweenthe locational marginal price and the nodal load is equivalent to therelationship between the dependent variables ω,r and the independentvariable b in the KKT condition. Therefore, “when the nodal loadchanges, the locational marginal price remains unchanged” is equivalentto “when the independent variable b changes, the dependent variables ω,rremain unchanged”.

It is supposed that in a base state c=c₀ and b=b₀, the dependentvariables in the KKT condition may be ω=ω₀,r=r₀,x*=x₀*. In order toensure that the dependent variables ω,r remain unchanged after theindependent variable b is superimposed by □b, it is necessary to ensurethat when the independent variables become c=c₀ and b=b₀+□b, thedependent variables in the KKT condition satisfy the following form:ω=ω₀, r=r₀, x*=x₀*+□x*.

Therefore, in the base state c=c₀, b=b₀, the KKT condition is asfollows:

$\quad\left\{ {\begin{matrix}{{A \cdot x_{0}^{*}} = b_{0}} \\{x_{0}^{*} \geq 0} \\{{{{A^{T} \cdot \omega_{0}} + r_{0}} = c^{T}},{r \geq 0}} \\{{{r_{0}}^{T}x_{0}^{*}} = 0}\end{matrix},} \right.$

When the independent variables change in the base state, that is, c=c₀,b=b₀+□b, the KKT condition is as follows:

$\left\{ {\begin{matrix}{{A \cdot \left( {x_{0}^{*} + {\bullet\; x^{*}}} \right)} = {b_{0} + {\bullet b}}} \\{{x_{0}^{*} + {\bullet x^{*}}} \geq 0} \\{{{{A^{T} \cdot \omega_{0}} + r_{0}} = c^{T}},{r_{0} \geq 0}} \\{{{r_{0}}^{T}\left( {x_{0}^{*} + {\bullet\; x^{*}}} \right)} = 0}\end{matrix}.} \right.$

The expansion equation of the first-order KKT condition in theincremental form may be derived from the above equation as follows:

$\left\{ {\begin{matrix}{{{A \cdot \bullet}\; x^{*}} = {\bullet\; b}} \\{{{{r_{0}}^{T} \cdot \bullet}\; x^{*}} = 0} \\{{x_{0} + {\bullet\; x^{*}}} \geq 0}\end{matrix}.} \right.$

(2-2) A projection matrix is designed to derive the parameter changingdomain of loads under the case that guarantees the constant locationalmarginal price:

The projection matrix P of an equation r₀ ^(T)·□x*=0 is defined asfollows:

P=I−r ₀·(r ₀ ^(T) ·r ₀)⁻¹ ·r ₀.

The expansion equation of the first-order KKT condition in theincremental form may be transformed into the parameter changing domainof loads under the case that guarantees the constant locational marginalprice as follows:

S□{A·P·□y|x*+P□y≥0, P=I−r ₀·(r ₀ ^(T) ·r ₀)⁻¹ ·r ₀ ^(T) , □y∈R ^(n)}.

The method for calculating the parameter changing domain of loads in thepower system under the case that guarantees the constant locationalmarginal price in the electricity market, provided in the disclosure,may have the following advantages.

What is claimed is:
 1. A method for calculating a parameter changingdomain of loads in a power system under a case that guarantees aconstant locational marginal price in an electricity market, comprising:(1) establishing a clearing model on the locational marginal price in ageneral form, comprising: (1-1) establishing the clearing model on thelocational marginal price as follows:$\mspace{79mu}{\min\limits_{p_{i}}{\sum\limits_{i = 1}^{Ng}\;{c_{i}p_{i}}}}$$\mspace{79mu}{{{satisfyin}\text{g:}\mspace{14mu}{\sum\limits_{i = 1}^{Ng}\; p_{i}}} = {\sum\limits_{j = 1}^{N}\; D_{j}}}$${P_{i}^{\min} \leq p_{i} \leq P_{i}^{\max}},{\forall{i \in {{N_{g} - F_{l}^{\max}} \leq {{\sum\limits_{i = 1}^{Ng}\;{S_{l.i}p_{i}}} - {\sum\limits_{j = 1}^{N}\;{S_{l.j}D_{j}}}} \leq F_{l}^{\max}}}},{\forall{l \in L}},$where, N_(g) represents a number of generator nodes in the power system;N_(g) represents a set of serial numbers of the generator nodes,N_(g)□{1, 2, . . . , N_(g)}; N represents a total number of nodes in thepower system; L represents a set of serial numbers of branches in thepower system, L□{1,2, . . . , L}, L represents a total number of thebranches in the power system; p_(i),i∈N_(g) represents a power variableof generator i; c_(i),i∈N_(g) represents a power cost coefficient ofgenerator i; P_(i) ^(max) represents an upper power limit of thegenerator node; P_(i) ^(min) represents a lower power limit of thegenerator node; D_(j),j∈{1,2, . . . , N} represents a nodal load of thepower system; F_(l) ^(max),l∈L represents a capacity of branch l; andS_(l,i), l∈L,i∈{1, 2, . . . , N} represents a power transferdistribution factor; there is a following linear relationship betweenlocational marginal prices and Lagrangian multipliers of constraints:Λ=τ−S _(LN) ^(T)(μ^(L,max)−μ^(L,min)), where, Λ represents a columnvector including locational marginal prices of the nodes in the powersystem in an order of serial numbers of the nodes; τ represents aLagrangian multiplier for the power balance constraint${{\sum\limits_{i = 1}^{Ng}p_{i}} = {\sum\limits_{i = 1}^{N}D_{i}}};$μ^(L,max) represents a column vector including Lagrangian multipliersfor the upper bound constraint${{\overset{Ng}{\sum\limits_{i = 1}}{S_{l.i}p_{i}}} - {\sum\limits_{i = 1}^{N}{S_{l.i}D_{i}}}} \leq F_{l}^{\max}$in the power system in an order of serial numbers of the branches;μ^(L,min) represents a column vector including Lagrangian multipliersfor the lower bound constraint${- F_{l}^{\max}} \leq {{\sum\limits_{i = 1}^{Ng}{S_{l.i}p_{i}}} - {\sum\limits_{i = 1}^{N}{S_{l.i}D_{i}}}}$in the power system in an order of serial numbers of the branches;S_(LN) represents a power transfer distribution factor matrix; and Trepresents a matrix transpose; (1-2) letting a variablep_(i)′=p_(i)−P_(i) ^(min) to transform a decision variable p_(i) of theclearing model into a pure non-negative variable p_(i)′, and introducingslack variables p_(i) ^(sl),f_(l) ^(sl,min),f_(l) ^(sl,max) to transformthe clearing model into a linear programming in a general form asfollows:$\mspace{79mu}{\min\limits_{p_{i}}{\sum\limits_{i = 1}^{Ng}{c_{i}p_{i}^{\prime}}}}$$\mspace{79mu}{{{satisfyin}\text{g:}\mspace{14mu}{\sum\limits_{i = 1}^{Ng}p_{i}^{\prime}}} = {{\sum\limits_{i = 1}^{N}D_{i}} - {\sum\limits_{i = 1}^{Ng}P_{i}^{\min}}}}$${{p_{i}^{\prime} + p_{i}^{sl}} = {P_{i}^{\max} - P_{i}^{\min}}},{{\forall{i \in {N_{g} - F_{l}^{\max}}}} = {{\sum\limits_{i = 1}^{Ng}{S_{l.i}p_{i}^{\prime}}} + {\sum\limits_{i = 1}^{Ng}{S_{l.i}P_{i}^{\min}}} - {\sum\limits_{i = 1}^{N}{S_{l.i}D_{i}}} - f_{l}^{{sl},\min}}},{\forall{l \in L}}$$\mspace{79mu}{{{{\sum\limits_{i = 1}^{Ng}{S_{l.i}p_{i}^{\prime}}} + {\sum\limits_{i = 1}^{Ng}{S_{l.i}P_{i}^{\min}}} - {\sum\limits_{i = 1}^{N}{S_{l.i}D_{i}}} + f_{l}^{{sl},\max}} = F_{l}^{\max}},{\forall{l \in L}}}$     p_(i)^(′), p_(i)^(sl), f_(l)^(sl, min ), f_(l)^(sl, max ) ≥ 0;(1-3) simplifying the clearing model obtained in (1-2) into the linearprogramming in the general form, as follows:$\min\limits_{x}{\sum{c \cdot x}}$ satisfying:  A ⋅ x = b, x ≥ 0 where,matrix A and vectors c,b correspond to parameters of the clearing modelas follows: ${A = \begin{bmatrix}e_{G} & \; & \; & \; \\I_{G} & I_{G} & \; & \; \\S_{LG} & \; & {­I_{L}} & \; \\S_{LG} & \; & \; & I_{L}\end{bmatrix}},{x = \begin{bmatrix}p^{\prime} \\p^{sl} \\f^{{sl},\min} \\f^{{sl},\max}\end{bmatrix}},{b = \begin{bmatrix}{{e_{D}D} - {e_{G}P^{\min}}} \\{P^{\max} - P^{\min}} \\{{S_{LN}D} - {S_{LG}P^{\min}} - F^{\max}} \\{{S_{LN}D} - {S_{LG}P^{\min}} + F^{\max}}\end{bmatrix}},{c = \begin{bmatrix}c^{T} & 0 & 0 & 0\end{bmatrix}}$ where, e_(G) is a matrix whose elements of dimension1×N_(g) are all 1; e_(D) is a matrix whose elements of dimension 1×N areall 1; I_(G) is a unit matrix with dimension N_(g)×N_(g); I_(L) is aunit matrix with dimension L×L; S_(LG) is a sub-matrix formed by columnscorresponding to the generator nodes; (2) deriving and calculating theparameter changing domain of loads under the case that guarantees theconstant locational marginal price based on a first-order KKT conditionof the clearing model in (1-3) in the general form, comprising: (2-1)deriving the first-order KKT condition in an incremental form asfollows: $\left\{ {\begin{matrix}{{A \cdot x^{*}} = b} \\{x^{*} \geq 0} \\{{{{A^{T} \cdot \omega} + r} = c^{T}},{r \geq 0}} \\{{r^{T} \cdot x^{*}} = 0}\end{matrix},} \right.$ where, ω is a Lagrangian multiplier vector ofthe constraint condition A·x=b; r is a Lagrangian multiplier vector ofthe constraint condition x≥0; c,b are independent variables in the KKTcondition; ω,r,x* are dependent variables in the KKT condition;supposing that in a base state c=c₀ and b b₀, the dependent variables inthe KKT condition being ω=ω₀,r=r₀,x*=x₀*; wherein in order to ensurethat the dependent variables ω,r remain unchanged after the independentvariable b is superimposed by □b, it is necessary to ensure that whenthe independent variables become c=c₀ and b=b₀+□b, the dependentvariables in the KKT condition satisfy the following form: ω=ω₀, r=r₀,x*=x₀*+□x*; in the base state c=c₀, b=b₀, the KKT condition is asfollows: $\left\{ {\begin{matrix}{{A \cdot x_{0}^{*}} = b_{0}} \\{x_{0}^{*} \geq 0} \\{{{{A^{T} \cdot \omega_{0}} + r_{0}} = c^{T}},{r \geq 0}} \\{{{r_{0}}^{T}x_{0}^{*}} = 0}\end{matrix},} \right.$ when the independent variables change in thebase state, the KKT condition is as follows: $\left\{ {\begin{matrix}{{A \cdot \left( {x_{0}^{*} + {\bullet x^{*}}} \right)} = {b_{0} + {\bullet\; b}}} \\{{x_{0}^{*} + {\bullet x^{*}}} \geq 0} \\{{{{A^{T} \cdot \omega_{0}} + r_{0}} = c^{T}},{r_{0} \geq 0}} \\{{{r_{0}}^{T}\left( {x_{0}^{*} + {\bullet\; x^{*}}} \right)} = 0}\end{matrix},} \right.$ an expansion equation of the first-order KKTcondition in the incremental form is derived from the above equation asfollows: $\left\{ {\begin{matrix}{{{A \cdot \bullet}\; x^{*}} = {\bullet\; b}} \\{{{{r_{0}}^{T} \cdot \bullet}\; x^{*}} = 0} \\{{x_{0} + {\bullet\; x^{*}}} \geq 0}\end{matrix};} \right.$ (2-2) design a projection matrix to derive theparameter changing domain of loads under the case that guarantees theconstant locational marginal price: defining the projection matrix P ofan equation r₀ ^(T)·□x*=0 as follows:P=I−r ₀·(r ₀ ^(T) ·r ₀)⁻¹ ·r ₀ ^(T); the expansion equation of thefirst-order KKT condition in the incremental form is transformed intothe parameter changing domain of loads under the case that guaranteesthe constant locational marginal price as follows:S□{A·P·□y|x*+P□y≥0, P=I−r ₀·(r ₀ ^(T) ·r ₀)⁻¹ r ₀ ^(T) , □y∈R ^(n)}.